Foam Flow Experiments. I. Estimation of the Bubble generationCoalescence Function
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Abstract
Gas injection leads to foam formation in porous media in the presence of surfactant solutions, which is used for flow diversion and enhanced oil recovery. We present here laboratory experiments of coinjecting nitrogen and sodium \(C_{1416}\) alpha olefin sulfonate with two concentrations: \(20 \times \) the critical micelle concentration (CMC) in an unconsolidated sandpack of 1860 Darcy and at the CMC for a Bentheimer sandstone of 3 Darcy. The steady state profile for the unconsolidated sandpack is achieved after 1.3 pore volumes, whereas for Bentheimer sandstone, steady state is obtained after injection of 12–15 pore volume. A model that leads to four equations, viz., a pressure equation, a water saturation equation, a bubble density equation and a surfactant transport–adsorption equation, is used to explain the experimental pressure drop. It is asserted that the experimental pressure drop across the measurement points can be used to obtain a first estimate of the average bubble density, which can be further used to obtain part of the source term in the bubble density equation. If we consider flowing fraction of foam, the rate of change of bubble density during transient state can be equated to the bubble density generationcoalescence function plus the terms accounted for bubble transport by convection and diffusion divided by porosity and saturation.
Keywords
Foam model Experiments Bubble generationcoalescence function Pressure drop history Adsorption1 Introduction
Foam can improve a water flood or a gas drive by decreasing the mobility (phase permeability/apparent viscosity) of the displacing fluids in the reservoir (Bond and Holbrook 1958; Craig and Lummus 1965; Holm 1968). The prediction of foam behavior in porous media relies on proper modeling of the mobility reduction validated by experiments. For the experiments where coinjection of gas and surfactant solution is considered, i.e., pregeneration of foam, one can use the saturation profiles, surfactant concentration profiles, the effluent water cut and the experimental pressure drop for the validation of models (Khatib et al. 1988; Falls et al. 1989; Persoff et al. 1991; Friedmann et al. 1991; Liu et al. 1992; Fergui et al. 1998; CarreteroCarralero et al. 2007; Du et al. 2011). A detailed summary of the literature on foam models can be found in Ma et al. (2014). Most modeling attempts are for experiments with surfactant concentrations well above the critical micelle concentration (CMC) where the buildup of pressure profiles occurs before the injection of about one pore volume. There are only a few experimental data reported in the literature of measured pressure drops with injected concentrations around the CMC, for example by Apaydin and Kovscek (2001). The effluent concentration profile of Suntech IV (an alkyl toluene sulfonate) for Berea sandstone indicates a retardation factor of about 12 for 0.02 w/w % (Huh and Handy 1989). In case of Chaser CD1040 (an alpha olefin sulfonate) with concentrations near the CMC, the steady pressure drop profile is attained after 3–6 pore volumes (PV) in case of Berea sandstones (Chou 1991). To interpret such an observed delay in the pressure drop, one needs models that incorporate the transient development of foam bubbles. Therefore, our interest is in bubble population models by Falls et al. (1988), Ettinger and Radke (1992), Kovscek et al. (1995), Zitha et al. (2006) and Simjoo (2012) that can explain the transient pressure drop at high concentrations as well as at low concentrations, i.e., around CMC.
To validate our procedure, we carried out foam flow experiments that used coinjection of AOS and \(N_2\) at two concentrations, i.e., at 0.0375 w/w % in double distilled water (CMC) for Bentheimer and at 0.075 w/w % in brine (\(20 \times \hbox { CMC}\)) for the unconsolidated sandpack. We intend to use Bentheimer experiments for a comparison with experiments where nanoparticles were coinjected. The optimal stability of these particles was found at pH 3. Therefore, we choose pH = 3 for the Bentheimer case. Section 2.1 describes the experimental setup, sample preparation and measurement techniques. In contrast to most literature results, we describe foam flow experiments for a long time (maximum 30 pore volumes) in Sect. 2.2. In addition, we describe in Sect. 2.3 a separate singlephase adsorption test with a Bentheimer core at conditions identical to foam flow experiments to get the adsorption parameters. The main experimental result, shown in Sect. 2.4, is the pressure drop history between two measurement points and the time required to attain a stationary value. Section 3 is about modeling where Sect. 3.1 describes the 1D model considering downward vertical flow. The balance equations for waterfoamed gas and for bubble density with a bubble generationcoalescence function are described in Sect. 3.2. In addition, the pressure equation is used to calculate the simulated pressure drop. In the same Sect. 3.2, two model equations are included for the surfactant adsorption and transport. The procedure to estimate \(R(n_f)\) from the measured pressure drop in terms of the bubble density is explained in Sect. 3.3. The procedure to estimate \(\alpha \), a fitting parameter in the viscosity Eq. 2 from the surfactant concentration, is explained in Sect. 3.4. Boundary conditions are described in Sect. 3.5. We take the case of Bentheimer with the low concentration for the numerical simulation. The model equations from Sect. 3.2 are converted into weak form (Haberman 2004) in Sect. 3.6 to facilitate implementation in COMSOL, a commercial finite element software package. Subsequently, we describe the simulation results (Sect. 3.7) in terms of the water saturation and flowing fraction of foam. In addition, we describe the relation between the bubble density and surfactant concentration for the given simulation. Further, instead of splitting \(R(n_{f})\) like in most studies, we investigate terms on the other side of the bubble density equation, i.e., accumulation, convection and dispersion (diffusion). We determine the relative importance of the bubble accumulation and convectiondiffusion terms in Sect. 3.8, where the flowing fraction of foam and the foam saturation is considered. In Sect. 3.9, we compare the experimental pressure drop and the simulated pressure drop for the case of Bentheimer. We end with some conclusions about the procedure used, about the foam generationcoalescence function and about the estimate of the experimental pressure drop.
2 Experimental
2.1 Experimental SetUp
2.1.1 Porous Media and Solutions
Two different porous media are used in the experiments, viz., an unconsolidated sand of mean size of approximately one mm or Bentheimer cores. The pore size distribution for the unconsolidated sand as shown in Fig. 3 is obtained from image analysis, using an optical microscope. Prior to its use, the sand is treated with a potassium dichromate–sulfuric acid solution to make it completely waterwet. It is kept in the acid for one day and rinsed with tap water until all the acid is removed according to the procedure mentioned by Furniss et al. (1989). Subsequently, the sand is dried and poured, using the procedure of the seven sieves by Wygal (1963), in an acrylic transparent vertical tube to which we refer as the sandpack. The sandpack has an internal diameter of 0.039 m and a length of 0.15 m. The porosity of the unconsolidated sandpack is considered to be 0.38 (Panda and Lake 1994). In case of Bentheimer, the core for the experiment is cut from larger samples and not pretreated prior to its usage. The core is 17 cm in height and 4 cm in diameter and fitted in an aluminum core holder. The porosity of the Bentheimer core is measured to be 0.21 by comparing its weight with and without water, avoiding the presence of air bubbles. The permeability of the sandpack and the core is \(1860\pm 100\) Darcy and for Bentheimer is \(3\pm 0.5\) Darcy, respectively, measured by singlephase (water) test prior to foam flow experiments. We used 39.1 w/w % BIOTERGE\(^{\textregistered }\) AS40 sodium \(C_{1416}\) alpha olefin sulfonate (AOS) to prepare 0.3 w/w % AOS solution in 0.5 M brine and an AOS solution of 0.3 w/w% in acidic (pH = 3) water. We diluted both solutions for the foam experiments to prepare a 0.075 w/w % AOS solution in 0.5 M brine for the unconsolidated sandpack and 0.0375 w/w % AOS in acidic water with dissolved HCl (pH = 3) for Bentheimer. In addition, we measured the surface tension of the given surfactant in DD water in the presence of air with the du Nouy ring tensiometer. The critical micelle concentration of the surfactant decreases with the increase in the salt concentration. In 0.5 M brine, i.e., 30 g of NaCl in 1000 ml water, the critical micelle concentration of AOS is \( 4{\times } 10^{3} \,\mathrm{wt\%}\) as mentioned by Simjoo et al. (2013). From our surface tension study as shown in Fig. 6, in DD water the critical micelle concentration of AOS is 0.0375 w/w % (1.19 mmol/l). Therefore, the concentrations used for the experiments are \(20 \times \hbox { CMC}\) for the unconsolidated sandpack and CMC for Bentheimer.
2.1.2 Measurement of Pressure, Mass Flow and Temperature
2.2 Flow Experiments
We conducted ten foam flow experiments of which we report two here, i.e., with the unconsolidated sandpack and the Bentheimer sandstone core. As the goal of this paper is to show the procedure of extracting parameters for the bubble generationcoalescence function, we chose only these two experiments for reasons of concise presentation. For the unconsolidated sandpack, the flow is from bottom to top while for the Bentheimer core the flow is from top to the bottom. The foam experiments with unconsolidated sandpack are started at t \(=\) 0 s by flushing surfactant solution with concentrations of 0.075 w/w % AOS (\(20 \times \hbox { CMC}\)) at a rate of \(1.26 \times 10^{5}\hbox { m/s}\). After one PV of AOS, we achieved a singlephase steady pressure drop of 35,500 Pa/m between the measurement points. At t \(=\) 3309 s from the starting of the experiment, the back pressure valve is opened causing the backpressure to drop from 8.51 to 1.30 barA (absolute pressure). The injection pressure drops from 8.37 to 2.22 barA. At t \(=\) 3346 s (37 s after the release of the back pressure), \(N_{2}\) gas is injected at a rate of \(1.20 \times 10^{5}\hbox { m/s}\) in the already flowing AOS solution at the Tjunction between valve (3) and (4), 30 cm upstream of the sandpack inlet. As the foam develops inside the sandpack, the pressure at the injection point increases leading to compression of gas affecting the ratio of gas to total flow, i.e., foam quality. We calculated this foamed gas velocity or foam velocity (\(u_f\)) by dividing the injection foamed gas velocity with pressure ratio at the injection side. For the experiment with the Bentheimer core, we maintained 4 barA backpressure throughout the experiment. Before the experiment is started, we flushed \({\hbox {CO}}_{2}\) for 5 min followed by 100 ml of water with HCl (pH 3) to remove any trapped gas. The measurements for the foam experiment is started at t \(=\) 0 s by flushing a surfactant solution of 0.0375 w/w % concentration (\(\approx \) CMC) at a rate of \(3.75 \times 10^{5}\hbox { m/s}\). We waited to achieve a steady liquid pressure drop of 15,300 Pa/m between measuring points. At t \(=\) 3610 s (after 94 ml of AOS solution passed into the core), \(N_{2}\) gas is injected in the already flowing AOS solution at a flow velocity of \(7.8 \times 10^{5}\hbox { m/s}\). The inlet and outlet pressures at the time of gas injection were 4.22 and 4.18 barA, respectively. After injection of approximately 600 ml of AOS solution with a corresponding amount of gas, the measurements were stopped by closing the gas and liquid flow. The measured temperature fluctuated between 16 and \(17\,^{\circ }\mathrm{C}\). Table 1 summarizes the experiments mentioned above.
2.3 Adsorption Test
Summary of the foam flow experiments for Sandpack and Bentheimer
Porous media  Solvent  AOS w/w %  BP barA  \(U_\mathrm{{inj}}^\mathrm{{atm}} \times 10^{5}\) m/s  \(U_{f} \times 10^{5}\) m/s  \(U_{w} \times 10^{5}\) m/s  \(\Delta P \times 10^{5}\) Pa/m 

Sandpack  3 w/w% Brine  0.075  Atm.  1.20  1.01  1.25  \(22.0\pm 0.07\) 
Bentheimer  Acidic (pH 3)  0.0375  4  7.88  1.97  3.76  \(23.0\pm 0.2\) 
2.4 Experimental Results
3 Modeling
3.1 Physical Model
3.2 Model Equations
3.3 Rough Estimation of Bubble Density, \(n_{f}\), and the Source Term, \(R\left( n_{f}\right) \)
For the estimation of the bubble density, we follow in essence the bubble population approach adopted by Kovscek et al. (1995) for multiphase flow.
3.4 Estimation of Viscosity Coefficient \(\alpha \) from Surfactant Concentration
3.5 Boundary Conditions
Here we define boundary conditions based on experiments to be used in the simulations.
Surfactant concentration: We applied Dirichet boundary condition at the inlet with the prescribed value of C as \(C_\mathrm{{init}} + (C_\mathrm{{bound}}C_\mathrm{{init}})r(t)\) where \(C_\mathrm{{init}}\) is initial surfactant concentration and \({C_\mathrm{{bound}}}\) is produced surfactant concentration. \(C= {C_\mathrm{{init}}}\) at \(t=0\) (at the time of gas injection) for all x, \(C={C_\mathrm{{init}}}\) at \(x=0\) for all t, \(C={C_\mathrm{{bound}}}\) at \(x=L\) for all t. During the foam experiment with Bentheimer, 3 PV of surfactant solution was injected before gas coinjection. From the adsorption experiment, the adsorbed surface concentration \(C_s\) after 3 PV of surfactant is \(3 \times 10^{6}\hbox { mmol/m}^2\) at t \(=\) 0 for all x.
3.6 Numerical Solution
We consider here the Bentheimer experiment with low concentration (\(\approx \) CMC) with varying \(\alpha \). The viscosity coefficient \(\alpha \) is calculated from Eq. 25 with changing surfactant concentration, which itself is estimated from the adsorption test. We used the four 1D equations from Sect. 3.2 in their weak form (Haberman 2004) along coordinate x, i.e., the water saturation equation (Eq. 8), the pressure equation (Eq. 12), the bubble density equation (Eq. 13) and surfactant transport–adsorption equation (Eq. 15). We implemented the model in the multiphysics module of the commercial finite element software, COMSOL version 5.0. We consider a 1D geometry, consisting of a single domain with a length of 17 cm for the Bentheimer core. The quadratic Lagrangian elements are used with an element size of 0.00017 m for the Bentheimer core. A timedependent solver (generalized alpha) is used with a linear predictor and an amplification for high frequency of 0.75 (Jansen et al. 2000). In addition, we split the accumulation term of the bubble density and saturation product Eq. (19) in COMSOL into a sum containing a saturation derivative and a bubble density derivative. The termination technique was based on a prescribed tolerance, i.e., sum of absolute error (for each dependent variable) and relative error with maximum number of iterations 5. The convergence criterion for the solution was to arrive at a solution, which is within the specified tolerance. The step is accepted if the solver’s estimate of the (local) absolute error in the solution committed during a time step is smaller than the sum of absolute and relative error.
Parameters used for implementation of the model in the commercial software
Notation  Units  Description  Bentheimer 

Fluid properties  
\(\alpha \)  (\(10^{7}\)) \(\hbox {Ns}^{2/3}/\hbox {m}^{4/3}\)  Estimated viscosity coefficient  4.73–4.39 
\(\mu _w\)  (\(10^{5}\)) \(\hbox {N s/m}^2\)  Water viscosity  100 
\(\mu _g\)  (\(10^{5}\)) \(\hbox {N s/m}^2\)  Gas viscosity  8.5 
\(C_\mathrm{{AOS}}\)  () mmol/l  Surfactant conc.  1.19 
\(M_\mathrm{{AOS}}\)  g/mmol  Molecular weight of AOS  0.315 
\(C_\mathrm{{CMC}}\)  () mmol/l  Critical micelle concentration  1.19 
pH  (\(\))  pH of the solution  3.30 
\(u_\mathrm{{inj}}^\mathrm{{atm}}\)  \((10^{5})\) m/s  Injected Gas velocity  7.88 
\(u_{f}\)  \((10^{5})\) m/s  Foam velocity  1.97 
\(u_{w}\)  \((10^{5})\) m/s  Injected Liquid velocity  3.76 
\(\eta \)  \(\)  Foam Quality (\(u_f/(u_w+u_f\)))  0.33 
\(D_\mathrm{{cap}}\)  \((10^{7}\)) \(\hbox {m}^{2}/\hbox {s}\)  Capillary diffusion coefficient  9.30 
\(D_{n_{f}}\)  (\(10^{5}\)) \(\hbox {m}^{2}/\hbox {s}\)  Bubble diffusion  2.20 
\(n_\mathrm{{inj}}\)  /m  Injection bubbles density  231.00 
\(n_\mathrm{{init}}\)  /m  Initial bubbles density  231.00 
\(n_\mathrm{{inf}}\)  /m  Maximum bubbles density  47180.00 
\(D_{s}\)  (\(10^{7}\)) \(\hbox {m}^{2}/\hbox {s}\)  Surfactant diffusion  2.00 
\(C_{s}\)  (\(10^{5}\)) \(\hbox {mmol/m}^{2}\)  Initial Surfactant adsorbed  0.30 
\(S_\mathrm{{init}}\)  Initial water saturation  0.99  
Porous media properties  
\(\varphi \)  \(\)  Porosity  0.21 
\(\lambda \)  \(\)  Pore size distribution index  5.00 
k  (\(10^{12}\)) \(\hbox {m}^{2}\)  Permeability  3.00 
L  m  Length of the core  0.17 
R  (\(10^{5}\)) m  Capillary radius  1.10 
\(A_{s}\)  (\(10^{6}\)) \(\hbox {m}^{2}/\hbox {m}^{3}\)  Rock interstitial area  20.00 
\(p_\mathrm{{exit}}\)  barA  Pressure at the exit  4.10 
\(Q_{s}\)  (\(10^{5}\)) \(\hbox {m}^{3}/\hbox {gs}\)  Maximum adsorption capacity  14.28 
\(k_{a}\)  (\(10^{5}\)) mg/gs  Surface adsorption parameter  5.00 
\(k_{d}\)  (\(10^{5}\)) /s  Surface desorption parameter  90.00 
3.7 Numerical Results
The main output of the foam flow simulation is the pressure drop, water saturation and bubble density. As our primary goal in this work was to relate the bubble generation function to the experimental pressure drop, a detailed convergence analysis as in Ames (1977) for such a nonlinear problem is considered outside the scope of current paper. In case of the Bentheimer core with low concentration, the surfactant concentration along with the surfactant adsorption is compared with the experimental results.
3.8 Terms Contributing to the Pressure Drop
3.9 Comparison Between Experimental and Simulation Results
The simulated pressure profile in case of the Bentheimer experiment in Fig. 5 mimics features observed in the experimental result: the delayed foam generation and decrease in the pressure drop before it reaches steady value of \(2.3\times 10^{6}\hbox { Pa/m}\). The simulation shows that the water saturation increases after the maximum pressure drop is achieved. The increase in water saturation causes a decrease in the pressure drop across the measurement points. However, the bubble density continues to increase in this part of the simulation. After 15 PV of AOS and gas injection, the simulated water saturation achieved a constant value of 0.62, leading to a steady value of the pressure drop \(2.3\times 10^{6}\hbox { Pa/m}\). The mean absolute error between theoretical and experimental pressure drop is \(1.06 \times 10^5\hbox { Pa/m}\), i.e., within 10 % of the experimental pressure drop. Possible reasons for the imperfect match between simulation based on the proposed theoretical procedure and the experimental results are lack of fitting profiles in case of the uncertainty in the estimation of (a) the bubble density \(n_f\) from the experimental pressure drop, (b) the dn/dt from n and (c) the adsorption parameters \(k_{a}, k_{d}\) from the adsorption experiment. In addition, the adsorption parameters are taken from the singlephase experiment where the available surface area for adsorption is less than the surface area for multiphase foam flow. The role of bubble diffusion is not very well understood. These issues could be addressed in future work. In addition, to use the above procedure in the field, a further upscaling step is required, for instance using homogenization (Salimi and Bruining 2011).
4 Conclusions

The experimental pressure drop across the measurement points can be used to obtain a first estimate of the average bubble density, which can be further used to obtain part of the source term.

We measured the experimental pressure drop for two surfactant concentrations (20 \(\times \) CMC and CMC) and for two permeabilities (1860 and 3 Darcy). For the unconsolidated sandpack experiment (1860 Darcy), the steady state profiles are achieved after 1.3 pore volume (PV) of AOS injection with a concentration 20 \(\times \) CMC. For Bentheimer sandstone (3 Darcy), the steady state pressure drop is achieved after injection of 12–15 pore volume of low concentrations of AOS surfactant (of the order of the critical micelle concentration (CMC)) due to adsorption behavior. The experimental pressure drop shows a weak maximum in the unconsolidated sandpack and a strong maximum in the Bentheimer core.

A model that leads to four equations, viz., a pressure equation, a water saturation equation, a bubble density equation and a surfactant transport–adsorption equation can describe the pressure drop during the foam flow experiments. The viscosity coefficient \(\alpha \) in the Hirasaki–Lawson equation is estimated from the surfactant concentration to relate the foam viscosity to the estimated bubble density. The trapped gas fraction and the water saturation can be taken into account as a function of bubble density.

For the Bentheimer case, simulations indicate that the maximum in the pressure drop corresponds to a minimum in the water saturation. With the assumption that all gas is foamed and foam is the only phase in the porous medium, it is asserted that the dependence between the source term and bubble density is approximately obtained. The difference between simulated and experimental pressure drop is within 10 %, which suggests that the first estimate of the bubble generationcoalescence function is of the right order of magnitude.

Instead of splitting the source term \(R(n_f)\), we investigate terms on the other side of the bubble density equation, i.e., accumulation, convection and dispersion (diffusion). The study gives us an idea about their individual contribution to \(R(n_f)\). As we approximated \(\alpha \) as a resistance per lamella in the capillary tube, the generationcoalescence function can only be obtained within a factor. If we consider water saturation (twophase flow) and flowing fraction of foam, the rate of change of bubble density during transient state can be equated to the bubble density generation function plus the terms accounted for bubble transport by convection and diffusion divided by porosity and saturation.
Notes
Acknowledgments
We would like to thank Erasmus Mundus—India scholarship program for the scholarship and Shell for financial support. We acknowledge numerous useful suggestions of Prof. Dr. W. R. Rossen. We thank Dr. R. Farajzadeh for initiating the project and his support in designing the experiments. We acknowledge technical support from Dietz laboratory.
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